Kister polynomials in composition, namely
F
B
X
i
x
i
F
i
B
X
i
X
j>i
x
i
x
j
X
m
r0
r
F
i,j
B
ÿ
x
i
ÿ x
j
r
X
i
X
j>i
X
k>j
x
i
x
j
x
k
v
s
ijk
sF
i,j,k
B
s
i,j,k4
where F
B
is a composition-dependent property and
the parameters v
s
ijk
are given by:
v
s
ijk
x
s
ÿ
1 ÿ x
i
ÿ x
j
ÿ x
k
3
5
For a ternary A±B±C system when the binary inter-
action parameters
ÿ
r
F
i,j
B
are limited to the second-
order (m = 1), equation (1) becomes:
F
B
x
A
F
A
B
x
B
F
B
B
x
C
F
C
B
x
A
x
B
0
F
A,B
B
ÿ
x
A
ÿ x
B
1
F
A,B
B
x
A
x
C
0
F
A,C
B
ÿ
x
A
ÿ x
C
1
F
A,C
B
x
B
x
C
0
F
B,C
B
ÿ
x
B
ÿ x
C
1
F
B,C
B
x
A
x
B
x
C
x
A
A
F
A,B,C
B
x
B
B
F
A,B,C
B
x
C
C
F
A,B,C
B
6
When applied to diusion, the concentration-depen-
dent properties F
B
represented this way are the acti-
vation energy for the mobility (F
B
=Q
B
) and, after
a suggestion from Jo
È
nsson [11], the logarithm of the
frequency factor for the mobility coecient, i.e.
F
B
ln
ÿ
M
0
B
Y
B
. If there are thermodynamic
data available for the system considered, it is poss-
ible to include both tracer diusivities and chemical
diusivities in an assessment procedure similar to
the one used in the well-known CALPHAD
technique [3]. The parameters in the equations for
Y
B
and Q
B
are then determined from a least-square
®t to the experimental information. This method-
ology has been applied successfully to several tern-
ary and higher-order systems (see, e.g.
Jo
È
nsson [11, 13] and Engstro
È
m and A
Ê
gren [14]).
3.2. Eect of ferromagnetic ordering
In order to take into account the eect of ferro-
magnetic ordering on the diusivities the following
expression was suggested by Jo
È
nsson [4, 15]
Q
k
Q
p
k
DQ
mag
k
7
where Q
k
is the activation energy for the mobility,
Q
p
k
is the activation energy for the mobility in the
paramagnetic state with no magnetic ordering and
Q
mag
k
is the contribution to the activation energy
from magnetic ordering. Jo
È
nsson suggested that this
was given by the following expression:
DQ
mag
k
a
mag
2
DH
6
Q
p
k
ÿ
1
RT
8
The term
mag
DH is the magnetic enthalpy. It may
be expressed, for example, by the model proposed
by Hillert and Jarl [16]. a
2
is a factor of proportion-
ality which was determined by Jo
È
nsson by a ®t to
experimental information. He was able to success-
fully represent both diusion of several substitu-
tional elements, for which a
2
=0.3 was
determined [9], and diusion of interstitial elements
such as carbon and nitrogen, for which he found
a
2
=1.8 and a
2
=0.6, respectively [15].
3.3. Eect of chemical ordering
In the present work we shall accept the above
models for diusivities in disordered systems as well
as the eect of ferromagnetic ordering. We will
assume that equation (3) still holds. The starting
point for modeling the eect of chemical ordering
on the diusion in the B2-ordered b.c.c. phase is
the suggestion from Girifalco [17] who wrote the
activation energy for diusion in the ordered state
as
Q
k
Q
dis
k
1 a
k
ÿ
S
ord
2
9
where Q
dis
k
is the activation energy for diusion in
the disordered state, a
k
can be regarded as an adjus-
table parameter and S
ord
is the long-range order
parameter de®ned as
S
ord
p
a
A
ÿ p
b
A
10
where p
a
A
and p
b
A
are the probabilities of ®nding an
A atom on an a and b site, respectively. Equation
(9) was suggested by Girifalco to hold for both
chemical ordering and magnetic ordering. In the
case of magnetic ordering the long-range order par-
ameter S
ord
is replaced by S
mag
, the ratio of the
spontaneous magnetization at T to that at 0 K.
Equation (9) was used by Ijima and Lee [18] for
representing the tracer diusivities of an Fe ±
50.8 at.% Co alloy. Girifalco's derivation was made
for the stoichiometric composition (x
A
=0.5). Tokei
et al. [19] extended the derivation of Girifalco to
the composition (x
A
=0.25) and found that
equation (9) still holds. This expression was then
used successfully to represent their experimental in-
formation. To the knowledge of the present
authors, Girifalco's model has so far only been
used for representing diusivities of unary and
binary systems. However, a natural generalization
of Girifalco's model to multicomponent systems
was recently suggested by A
Ê
gren [20], who wrote
the activation energy for diusion of a component
k in the ordered state as
Q
k
Q
dis
k
DQ
ord
k
11
with DQ
ord
k
as a contribution to the activation
energy given by
DQ
ord
k
X
i
X
i6j
DQ
order
kij
y
a
i
y
b
j
ÿ x
i
x
j
12
where y
a
i
is the site fraction of component i on the
HELANDER and A
Ê
GREN: A PHENOMENOLOGICAL TREATMENT 1143